When quantum mechanics is understood as a new generalized theory of probability - to be called the quantum probability theory - mysteries and controversies regarding quantum mechanics are dissolved. In the classical probability theory, that a measurement of some system requires an additional measurement apparatus is of insignificant importance - in the quantum probability theory, this comes to change. For one central single reason around a particular classical probability equation, the generalized probability view has not gained much traction, despite the fact that this essentially echoes (and provides logical underpinnings of) the conventional wisdom that `quantum mechanics just works as it is.' A classical probability axiom is just an initial intuition - there is no reason why we have to dogmatically cling onto axioms that can clearly be generalized. Issues with the principle of indifference in the classical probability theory are emphasized, along with the quantum reconstruction project of deriving quantum mechanics from epistemic requirements and potential quantum gravity consequences from the principle of maximum entropy.
When quantum mechanics is understood as a new generalized theory of probability - to be called the quantum probability theory - mysteries and controversies regarding quantum mechanics are dissolved. In the classical probability theory, that a measurement of some system requires an additional measurement apparatus is of insignificant importance - in the quantum probability theory, this comes to change. For one central single reason around a particular classical probability equation, the generalized probability view has not gained much traction, despite the fact that this essentially echoes (and provides logical underpinnings of) the conventional wisdom that `quantum mechanics just works as it is.' A classical probability axiom is just an initial intuition - there is no reason why we have to dogmatically cling onto axioms that can clearly be generalized. Issues with the principle of indifference in the classical probability theory are emphasized, along with the quantum reconstruction project of deriving quantum mechanics from epistemic requirements and potential quantum gravity consequences from the principle of maximum entropy.
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
In his entry on "Quantum Logic and Probability Theory" in the Stanford Encyclopedia of Philosophy, Alexander Wilce (2012) writes that "it is uncontroversial (though remarkable) the formal apparatus quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over the 'quantum logic' of projection operators on a Hilbert space." For a long time, Patrick Suppes has opposed this view (see, for example, the paper collected in Suppes and Zanotti (1996). Instead of changing the logic and moving from a Boolean algebra to a non-Boolean algebra, one can also 'save the phenomena' by weakening the axioms of probability theory and work instead with upper and lower probabilities. However, it is fair to say that despite Suppes' efforts upper and lower probabilities are not particularly popular in physics as well as in the foundations of physics, at least so far. Instead, quantum logic is booming again, especially since quantum information and computation became hot topics. Interestingly, however, imprecise probabilities are becoming more and more popular in formal epistemology as recent work by authors such as James Joye (2010) and Roger White (2010) demonstrates.
Abstract The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extra‐mathematical content. Here, a representative series of specifically information‐based treatments—from partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systems—is reviewed.
Abstract In the 1950s and 1960s it was causing interest among physicists that in the formalism of quantum mechanics (complex-valued) ‘probability amplitudes’ obeyed laws similar to the laws obeyed by probabilities in the ordinary probability calculus. But this did not then lead decisively to the claim that probabilities should be represented by complex numbers. It became fashionable instead to regard probability amplitudes as an abstract concept from which actual probabilities could be derived by taking the squared moduli of the amplitudes. In this monograph, however, we make another attempt to show how probability amplitudes might after all be identified with actual probabilities. To do this, probability itself is interpreted in a rational Bayesian manner (i.e. as an extension of logic) and a new (complex-valued) probability theory is formulated that incorporates the uncertainty principle (i.e. that takes account of the fact that acquisition of knowledge of a quantum mechanical process generally interferes with it). Taking this probability theory as the new logic of science, and assuming certain physical laws and properties of matter, an interpretation of non-relativistic quantum mechanics is built up. It is consistent with the usual quantum mechanical formalism but allows a clear distinction to be made between the physical world and our knowledge of it.
The topic of probability in quantum mechanics is rather vast. In this chapter it is discussed from the perspective of whether and in what sense quantum mechanics requires a generalization of the usual (Kolmogorovian) concept of probability. The focus is on the case of finite-dimensional quantum mechanics (which is analogous to that of discrete probability spaces), partly for simplicity and partly for ease of generalization. While the main emphasis is on formal aspects of quantum probability (in particular the non-existence of joint distributions for incompatible observables), the discussion relates also to notorious issues in the interpretation of quantum mechanics. Indeed, whether quantum probability can or cannot be ultimately reduced to classical probability connects rather nicely to the question of 'hidden variables' in quantum mechanics.
The EPR controversy between Albert Einstein and Niels Bohr about the completeness of the quantum mechanics is one of the most essential contemporary debates in physics with respect to the issue of realism. Initially considered as a though-experiment, the EPR paradox was meant to evidence the inadequacy of quantum mechanics with respect to the traditional leading principle of physics such as locality, independence of the system elements of reality with respect to the observer or semantic reference of theories to elements of the reality in itself. Bell’s mathematical inquiry opened the possibility to empirically discriminate between the view of Bohr and Einstein through the so-called Bell’s inequalities. Based on these inequalities, the experimental test provided in particular by the work of Alain Aspect settled the debate in favor of Bohr and the completeness of quantum mechanics. The previously evident assumption that physical theories are describing the reality in it-self can thus be questioned. Several approaches have been proposed to face that challenge, such as the orthodox quantum physics formulation coupled with the interpretation of Bernard d’Espagnat , the non-local theory of De Broglie-Bohm or the quantum mechanics approach with extended probability of James B. Hartle . In this paper, we consider the idea according to which the field of experiments and measurements on quantum systems is a research context where several epistemic values traditionally leading the scientific activity are in conflict. Instead of conceiving the transition from classical to quantum physics along Thomas Kuhns’ theory of revolutionary paradigm change , characterized by a strong incommensurability between the old and the new approaches, we argue that this epistemological shift could be rationally understood as an incremental evolution of the epistemic values leading scientific activity. As the analysis in this paper aims to show, such a rational reconstruction shows a process of gradual change, where core epistemic values are maintained, while others are broadened to account for the new scientific approaches. The settling of the EPR controversy could be understood as the indication that the previously assumed epistemic values cannot be consistently held together in the quantum context. The values of logical consistency and empirical adequacy remained universally shared, but the controversy leads to question the other traditionally admitted principles, such as the locality, the requirement of semantic references to the reality in itself and the independence of experimental results with respect to the observer, all tightly connected with the realistic conceptions of the world. This first part of the paper characterizes the selected contemporary approaches in quantum context with respect to the modification of the epistemic principles they adopt to restore their consistency. While the de Broglie-Bohm theory explicitly follows the requirement of referring to the reality in itself at the expense of the locality principle, the approaches of d’Espagnat and of Hartle give priority to the values of theoretical framework simplicity and predictive power. In the case of d’Espagnat, the descriptive purpose of the scientific activity remains, but only in a weak objectivity perspective, restricted to the “empirical reality” in which the observer is involved. In the case of Hartle, the extended probability formulation of quantum mechanics is conceived as “an extension of familiar probability theory that incorporates negative probabilities”6, which relies on the Bayesian conception of “probabilities as instruction for betting” (negative probability is thus associated to the instruction “don’t bet”). In a second part, the paper uses this reconstruction of the various theories in order to address the epistemic status of the principles of descriptive power and of independence of the observed system with respect to the observer. In particular, our analysis aims to assess if a reference to the observer is part of the reality description or if such reference is avoidable. This is of prior interest in exploring the main characteristics of a realistic approach that would be applicable in the context of experiments and measurements on quantum systems. In that perspective, we argue that the direct holistic realism, developed by Hilary Putnam in The Threefold Cord , could be a promising intermediate option between a classical realism, which assumes that theoretical structures are describing the reality in it-self, and a more instrumentalist view, focusing mainly on the modeling of the empirical reality as a result of experimental operations while rejecting access to the reality in itself.